Let $(X,\mathscr{H})$ and $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from $(X,\mathscr{H})$ to $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ is a continuous map from $X$ to $X^{\text{'}}$ which preserves the biharmonic structures of $X$ and $X^{\text{'}}$. In the present work we study this notion and characterize in some cases the biharmonic morphisms between $X$ and $X^{\text{'}}$ in terms of harmonic morphisms between the harmonic spaces associated with $(X,\mathscr{H})$ and $(X^{\text{'}},{\mathscr{H}}^{\text{'}})$ and the coupling kernels of them.